# Coverage for pygeodesy/triaxials.py : 95%

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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 | # -*- coding: utf-8 -*- u'''Triaxal ellipsoid classes L{JacobiConformal}, Jacobi's conformal projection, trancoded from I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/ doc/classGeographicLib_1_1JacobiConformal.html#details>} to pure Python, I{ordered} L{Triaxial} and I{unordered} L{Triaxial_} and miscellaneous classes L{BetaOmega2Tuple}, L{BetaOmega3Tuple}, L{Jacobi2Tuple} and L{TriaxialError}. @see: U{Geodesics on a triaxial ellipsoid<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# Geodesics_on_a_triaxial_ellipsoid>} and U{Triaxial coordinate systems and their geometrical interpretation<https://www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. ''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon # from pygeodesy.basics import isscalar, map1, _zip # from .fsums, .namedTuples, .streprs from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, _EPS2e4, INT0, PI2, PI_3, PI4, \ _0_0, _0_5, _1_0, _N_1_0, isfinite, isnear1, \ _4_0 # PYCHOK used! from pygeodesy.datums import Datum, Ellipsoid, _spherical_datum, _WGS84 # from pygeodesy.ellipsoids import Ellipsoid # from .datums # from pygeodesy.elliptic import Elliptic # ._MODS # from pygeodesy.errors import _ValueError # from .streprs from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, hypot2, hypot2_, norm2 from pygeodesy.fsums import Fsum, fsum, fsum_, isscalar, Property_RO from pygeodesy.interns import NN, _a_, _b_, _c_, _distant_, _height_, _inside_, \ _invalid_, _near_, _not_, _NOTEQUAL_, _null_, _opposite_, \ _outside_, _SPACE_, _spherical_, _too_, _x_, _y_ # from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d from pygeodesy.named import _NamedBase, _NamedTuple, _Pass from pygeodesy.namedTuples import LatLon3Tuple, map1, Vector3Tuple, Vector4Tuple # from pygeodesy.props import Property_RO # from .fsums from pygeodesy.streprs import Fmt, _ValueError, _zip from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, Radians, Radius from pygeodesy.utily import asin1, atan2d, km2m, sincos2, sincos2d, sincos2d_ from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d from math import atan2, fabs, sqrt __all__ = _ALL_LAZY.triaxials __version__ = '22.11.02' _not_ordered_ = _not_('ordered') _TRIPS = 256 # max 55, Eberly 1074? class _ToNamedBase(_NamedBase): '''(INTERNAL) C{-.toDegrees}, C{-.toRadians} base. ''' def _toDegrees(self, a, b, *c, **toDMS_kwds): if toDMS_kwds: toDMS = _MODS.dms.toDMS a = toDMS(a.toDegrees(), **toDMS_kwds) b = toDMS(b.toDegrees(), **toDMS_kwds) elif isinstance(a, Degrees) and \ isinstance(b, Degrees): return self else: a, b = a.toDegrees(), b.toDegrees() return self.classof(a, b, *c, name=self.name) def _toRadians(self, a, b, *c): return self if isinstance(a, Radians) and \ isinstance(b, Radians) else \ self.classof(a.toRadians(), b.toRadians(), *c, name=self.name) class BetaOmega2Tuple(_NamedTuple, _ToNamedBase): '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees}). ''' _Names_ = ('beta', 'omega') _Units_ = (_Pass, _Pass) def toDegrees(self, **toDMS_kwds): '''Convert this L{BetaOmega2Tuple} to C{Degrees} or C{toDMS}. @return: L{BetaOmega2Tuple}C{(beta, omega)} with C{beta} and C{omega} both in C{Degrees} or as an L{toDMS} string provided some B{C{toDMS_kwds}} are supplied. ''' return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) def toRadians(self): '''Convert this L{BetaOmega2Tuple} to C{Radians}. @return: L{BetaOmega2Tuple}C{(beta, omega)} with C{beta} and C{omega} both in C{Radians}. ''' return _ToNamedBase._toRadians(self, *self) class BetaOmega3Tuple(_NamedTuple, _ToNamedBase): '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees}) and the C{height}, rather the (signed) I{distance} to the triaxial's surface (measured along the radial line to the triaxial's center) in C{meter}, conventionally. ''' _Names_ = BetaOmega2Tuple._Names_ + (_height_,) _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) def toDegrees(self, **toDMS_kwds): '''Convert this L{BetaOmega3Tuple} to C{Degrees} or C{toDMS}. @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and C{omega} both in C{Degrees} or as an L{toDMS} string provided some B{C{toDMS_kwds}} are supplied. ''' return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) def toRadians(self): '''Convert this L{BetaOmega3Tuple} to C{Radians}. @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and C{omega} both in C{Radians}. ''' return _ToNamedBase._toRadians(self, *self) def to2Tuple(self): '''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}. ''' return BetaOmega2Tuple(*self[:2]) class Jacobi2Tuple(_NamedTuple, _ToNamedBase): '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} projection, both in C{Radians} (or C{Degrees}). ''' _Names_ = (_x_, _y_) _Units_ = (_Pass, _Pass) def toDegrees(self, **toDMS_kwds): '''Convert this L{Jacobi2Tuple} to C{Degrees} or C{toDMS}. @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} both in C{Degrees} or as an L{toDMS} string provided some B{C{toDMS_kwds}} are supplied. ''' return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) def toRadians(self): '''Convert this L{Jacobi2Tuple} to C{Radians}. @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} both in C{Radians}. ''' return _ToNamedBase._toRadians(self, *self) class Triaxial_(_NamedBase): # _NamedEnumItem '''I{Unordered} triaxial ellipsoid and base class. Triaxial ellipsoids with right-handed semi-axes C{a}, C{b} and C{c}, oriented such that the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0, while the small principal ellipse C{ac} is the prime meridian, plane I{Y}=0, I{omega}=0. The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie on the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}. @note: I{Geodetic} C{lat}- and C{lon}gitudes are in C{degrees}, I{geodetic} C{phi} and C{lam}bda are in C{radians}, but I{ellipsoidal} lat- and longitude C{beta} and C{omega} are in C{Radians} by default (or in C{Degrees} if converted). ''' _unordered = True def __init__(self, a_triax, b=None, c=None, name=NN): '''New I{unordered} L{Triaxial_}. @arg a_triax: C{X} semi-axis (C{scalar}, conventionally in C{meter}) or an other L{Triaxial} or L{Triaxial_} instance. @kwarg b: C{Y} semi-axis (C{meter}, same units as B{C{a}}), required if C{B{a_triax} is scalar}, ignored otherwise. @kwarg c: C{Z} semi-axis (C{meter}, same units as B{C{a}}), required if C{B{a_triax} is scalar}, ignored otherwise. @kwarg name: Optional name (C{str}). @raise TriaxialError: Invalid semi-axis or -axes. ''' try: a = a_triax t = a._abc3 if isinstance(a, Triaxial_) else map1(float, a, b, c) except (TypeError, ValueError) as x: raise TriaxialError(a=a, b=b, c=c, cause=x) if name: self.name = name a, b, c = self._abc3 = t if self._unordered: # == not isinstance(self, Triaxial) s, _, t = sorted(t) if not (isfinite(t) and s > 0): raise TriaxialError(a=a, b=b, c=c) # txt=_invalid_ elif not (isfinite(a) and a >= b >= c > 0): raise TriaxialError(a=a, b=b, c=c, txt=_not_ordered_) elif not (a > c and self._a2c2 > 0 and self.e2ac > 0): raise TriaxialError(a=a, c=c, e2ac=self.e2ac, txt=_spherical_) def __str__(self): return self.toStr() @Property_RO def a(self): '''Get the (largest) C{x} semi-axis (C{meter}, conventionally). ''' a, _, _ = self._abc3 return Radius(a=a) @Property_RO def _a2b2(self): '''(INTERNAL) Get C{a**2 - b**2} == E_sub_e**2. ''' a, b, _ = self._abc3 return ((a - b) * (a + b)) if a != b else _0_0 @Property_RO def _a2_b2(self): '''(INTERNAL) Get C{(a/b)**2}. ''' a, b, _ = self._abc3 return (a / b)**2 if a != b else _1_0 @Property_RO def _a2c2(self): '''(INTERNAL) Get C{a**2 - c**2} == E_sub_x**2. ''' a, _, c = self._abc3 return ((a - c) * (a + c)) if a != c else _0_0 @Property_RO def area(self): '''Get the surface area (C{meter} I{squared}). ''' c, b, a = sorted(self._abc3) if a > c: a = Triaxial(a, b, c).area if a > b else \ Ellipsoid(a, b=c).areax # a == b else: # a == c == b a = Meter2(area=a**2 * PI4) return a def area_p(self, p=1.6075): '''I{Approximate} the surface area (C{meter} I{squared}). @kwarg p: Exponent (C{scalar} > 0), 1.6 for near-spherical or 1.5849625007 for "near-flat" triaxials. @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}. ''' a, b, c = self._abc3 if a == b == c: a *= a else: _p = pow a = _p(fmean_(_p(a * b, p), _p(a * c, p), _p(b * c, p)), _1_0 / p) return Meter2(area_p=a * PI4) @Property_RO def b(self): '''Get the (middle) C{y} semi-axis (C{meter}, same units as B{C{a}}). ''' _, b, _ = self._abc3 return Radius(b=b) @Property_RO def _b2c2(self): '''(INTERNAL) Get C{b**2 - c**2} == E_sub_y**2. ''' _, b, c = self._abc3 return ((b - c) * (b + c)) if b != c else _0_0 @Property_RO def c(self): '''Get the (smallest) C{z} semi-axis (C{meter}, same units as B{C{a}}). ''' _, _, c = self._abc3 return Radius(c=c) @Property_RO def _c2_b2(self): '''(INTERNAL) Get C{(c/b)**2}. ''' _, b, c = self._abc3 return (c / b)**2 if b != c else _1_0 @Property_RO def e2ab(self): '''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b/a)**2}. ''' return Float(e2ab=(_1_0 - self._1e2ab) or _0_0) @Property_RO def _1e2ab(self): '''(INTERNAL) Get C{1 - e2ab} == C{(b/a)**2}. ''' a, b, _ = self._abc3 return (b / a)**2 if a != b else _1_0 @Property_RO def e2bc(self): '''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/b)**2}. ''' return Float(e2bc=(_1_0 - self._1e2bc) or _0_0) _1e2bc = _c2_b2 # C{1 - e2bc} == C{(c/b)**2} @Property_RO def e2ac(self): '''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/a)**2}. ''' return Float(e2ac=(_1_0 - self._1e2ac) or _0_0) @Property_RO def _1e2ac(self): '''(INTERNAL) Get C{1 - e2ac} == C{(c/a)**2}. ''' a, _, c = self._abc3 return (c / a)**2 if a != c else _1_0 @Property_RO def _Elliptic(self): '''(INTERNAL) Get class L{Elliptic} once. ''' return _MODS.elliptic.Elliptic def hartzell4(self, pov, los=None, name=NN): '''Compute the intersection of this triaxial's surface with a Line-Of-Sight from a Point-Of-View in space. @see: Function L{pygeodesy.hartzell4} for further details. ''' return hartzell4(pov, los=los, tri_biax=self, name=name) def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS): '''Compute the projection on and the height of a cartesian above or below this triaxial's surface. @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. @kwarg normal: If C{True} the projection is perpendicular to (the nearest point on) this triaxial's surface, otherwise the C{radial} line to this triaxial's center (C{bool}). @kwarg eps: Tolerance for root finding and validation (C{scalar}), use a negative value to skip validation. @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, C{y} and C{z} of the projection on or the intersection with and with the height C{h} above or below the triaxial's surface in C{meter}, conventionally. @raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no convergence in root finding or validation failed. @see: Method L{Ellipsoid.height4} and I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/Documentation/ DistancePointEllipseEllipsoid.pdf>}. ''' v = Vector3d(x_xyz, y, z) if isscalar(x_xyz) else _otherV3d(x_xyz=x_xyz) i, h = None, v.length if h < EPS0: # EPS x = y = z = _0_0 h -= min(self._abc3) elif self.isSpherical: r, _, _ = self._abc3 x, y, z = v.times(r / h).xyz h -= r elif normal: # perpendicular to triaxial a, b, c = self._abc3 x, y, z = v.xyz x, y, z, h, i = _normalTo5( x, y, z, self, eps=eps) if self.isOrdered else \ _normalTo5u(x, y, z, a,b,c, eps=eps) else: # radially to triaxial's center x, y, z = v.xyz x, y, z = self._radialTo3(z, hypot(x, y), y, x) h, i = v.minus_(x, y, z).length, None if h > 0 and self.sideOf(v, eps=EPS0) < 0: h = -h # below the surface return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__name__) @Property_RO def isOrdered(self): '''Is this triaxial I{ordered} and not I{spherical} (C{bool})? ''' a, b, c = self._abc3 return bool(a >= b > c) # b > c! @Property_RO def isSpherical(self): '''Is this triaxial I{spherical} (C{bool})? ''' a, b, c = self._abc3 return bool(a == b == c) def normal3d(self, x, y, z, length=_1_0): '''Get a 3-D vector at a cartesian on, perpendicular to this triaxial's surface. @arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). @arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). @kwarg length: Optional length and in-/outward direction (C{scalar}). @return: A C{Vector3d(x_, y_, z_)} normalized to B{C{length}}, pointing in- or outward for neg- respectively positive B{C{length}}. @note: Cartesian location C{(B{x}, B{y}, B{z})} must be on this triaxial's surface, use method L{Triaxial.sideOf} to validate. ''' # n = 2 * (x / a2, y / b2, z / c2) # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 # == unit(x, y / _1e2ab, z / _1e2ac).times(length) n = self._normal3d.times_(x, y, z) if n.length < EPS0: raise TriaxialError(x=x, y=y, z=z, txt=_null_) return n.times(length / n.length) @Property_RO def _normal3d(self): '''(INTERNAL) Get M{Vector3d((d/a)**2, (d/b)**2, (d/c)**2)}, M{d = max(a, b, c)}. ''' d = max(self._abc3) t = tuple(((d / x)**2 if x != d else _1_0) for x in self._abc3) return Vector3d(*t) def _norm2(self, s, c, *a): '''(INTERNAL) Normalize C{s} and C{c} iff not already. ''' if fabs(s) > _1_0 or fabs(c) > _1_0: # or \ \ # fabs(fsum(s**2, c**2, _N_1_0)) > EPS: s, c = norm2(s, c) if a: s, c = norm2(s * self.b, c * a[0]) return (s or _0_0), (c or _0_0) def _radialTo3(self, sbeta, cbeta, somega, comega): '''(INTERNAL) I{Unordered} helper for C{.height4}. ''' def _rphi(a, b, sphi, cphi): # <https://WikiPedia.org/wiki/Ellipse#Polar_form_relative_to_focus> # polar form: radius(phi) = a * b / hypot(a * sphi, b * cphi) return (b / hypot(sphi, b / a * cphi)) if a > b else ( (a / hypot(cphi, a / b * sphi)) if a < b else a) sa, ca = self._norm2(sbeta, cbeta) sb, cb = self._norm2(somega, comega) a, b, c = self._abc3 if a != b: a = _rphi(a, b, sb, cb) if a != c: c = _rphi(a, c, sa, ca) z, r = c * sa, c * ca x, y = r * cb, r * sb return x, y, z def sideOf(self, x_xyz, y=None, z=None, eps=EPS4): '''Is a cartesian above, below or on the surface of this triaxial? @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. @kwarg eps: Near surface tolerance(C{scalar}). @return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface within tolerance B{C{eps}}, otherwise a neg- or positive C{float} if in- respectively outside this triaxial. @see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}. ''' def _x2_a2_1(xyz, abc): for x, a in _zip(xyz, abc): # strict=True yield (x / a)**2 yield _N_1_0 t = (x_xyz, y, z) if isscalar(x_xyz) else _otherV3d(x_xyz=x_xyz).xyz s = fsum(_x2_a2_1(t, self._abc3), floats=True) return s if s > eps or s < -eps else INT0 def _sqrt(self, x): '''(INTERNAL) Helper. ''' if x < 0: raise TriaxialError(Fmt.PAREN(sqrt=x)) return _0_0 if x < EPS02 else sqrt(x) def toEllipsoid(self, name=NN): '''Convert this triaxial to an L{Ellipsoid}, provided C{a == b} or C{b == c}. @return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b}, this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}. @raise TriaxialError: This C{a} != C{b}, C{b} != C{c} and C{c} != C{a}. @see: Method L{Ellipsoid.toTriaxial}. ''' a, b, c = self._abc3 if a == b: # N = Z b = c elif b == c: # N = X a, b = b, a elif a != c: t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_) raise TriaxialError(a=a, b=b, c=c, txt=t) return Ellipsoid(a, b=b, name=name or self.name) def toStr(self, prec=9, name=NN, **unused): # PYCHOK signature '''Return this C{Triaxial} as a string. @kwarg prec: Precision, number of decimal digits (0..9). @kwarg name: Override name (C{str}) or C{None} to exclude this triaxial's name. @return: This C{Triaxial}'s attributes (C{str}). ''' T = Triaxial_ t = T.a, T.b, T.c, T.e2ab, T.e2bc, T.e2ac if isinstance(self, JacobiConformal): t += JacobiConformal.xyQ2, t += T.volume, T.area return self._instr(name, prec, props=t, area_p=self.area_p()) @Property_RO def volume(self): '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}. ''' a, b, c = self._abc3 return Meter3(volume=a * b * c * PI_3 * _4_0) class Triaxial(Triaxial_): '''I{Ordered} triaxial ellipsoid. @see: L{Triaxial_} for more information. ''' _unordered = False def __init__(self, a_triax, b=None, c=None, name=NN): '''New I{ordered} L{Triaxial}. @arg a_triax: Largest semi-axis (C{scalar}, conventionally in C{meter}) or an other L{Triaxial} or L{Triaxial_} instance. @kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required if C{B{a_triax} is scalar}, ignored otherwise. @kwarg c: Smallest semi-axis (C{meter}, same units as B{C{a}}), required if C{B{a_triax} is scalar}, ignored otherwise. @kwarg name: Optional name (C{str}). @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and must be ellipsoidal, C{B{a} > B{c}}. @raise TriaxialError: Semi-axes not ordered, spherical or invalid. ''' Triaxial_.__init__(self, a_triax, b=b, c=c, name=name) @Property_RO def _a2b2_a2c2(self): ''' @see: Method C{forwardBetaOmega}. ''' return self._a2b2 / self._a2c2 @Property_RO def area(self): '''Get the surface area (C{meter} I{squared}). @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}. ''' a, b, c = self._abc3 if a != b: kp2, k2 = self._k2_kp2 # swapped! aE = self._Elliptic(k2, _0_0, kp2, _1_0) c2 = self._1e2ac # cos(phi)**2 == (c/a)**2 s2 = self. e2ac # sin(phi)**2 == 1 - c2 s = sqrt(s2) r = asin1(s) # phi == atan2(sqrt(c2), s) b *= fsum_(aE.fE(r) * s, aE.fF(r) * c2 / s, c / a * c / b, floats=True) a = Meter2(area=a * b * PI2) else: # a == b > c a = Ellipsoid(a, b=c).areax return a def _exyz3(self, u): '''(INTERNAL) Helper for C{.forwardBetOmg}. ''' if u > 0: u2 = u**2 x = self._sqrt(_1_0 + self._a2c2 / u2) * u y = self._sqrt(_1_0 + self._b2c2 / u2) * u else: x = y = u = _0_0 return x, y, u def forwardBetaOmega(self, beta, omega, height=0, name=NN): '''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega} and height to cartesian. @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). @arg height: Height above or below the ellipsoid's surface (C{meter}, same units as this triaxial's C{a}, C{b} and C{c} semi-axes). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Method L{Triaxial.reverseBetaOmega} and U{Expressions (23-25)<https:// www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. ''' if height: h = Height_(height=height, low=-self.c, Error=TriaxialError) x, y, z = self._exyz3(h + self.c) else: x, y, z = self._abc3 # == self._exyz3(self.c) if z: # and x and y: sa, ca = _SinCos2(beta) sb, cb = _SinCos2(omega) r = self._a2b2_a2c2 x *= cb * self._sqrt(ca**2 + r * sa**2) y *= ca * sb z *= sa * self._sqrt(_1_0 - r * cb**2) return Vector3Tuple(x, y, z, name=name) def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, name=NN): '''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega} to cartesian coordinates I{on the triaxial's surface}. @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)} on the surface. @raise TriaxialError: This triaxial is near-spherical. @see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https:// www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. ''' t = self._radialTo3(sbeta, cbeta, somega, comega) return Vector3Tuple(*t, name=name) def forwardCartesian(self, x, y, z, name=NN, **normal_eps): '''Project a cartesian on this triaxial. @arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). @arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). @kwarg name: Optional name (C{str}). @kwarg normal_eps: Optional keyword arguments C{B{normal}=True} and C{B{eps}=EPS}, see method L{Triaxial.height4}. @see: Method L{Triaxial.height4} for further information and method L{Triaxial.reverseCartesian} to reverse the projection. ''' t = self.height4(x, y, z, **normal_eps) _ = t.rename(name) return t def forwardLatLon(self, lat, lon, height=0, name=NN): '''Convert I{geodetic} lat-, longitude and heigth to cartesian. @arg lat: Geodetic latitude (C{degrees}). @arg lon: Geodetic longitude (C{degrees}). @arg height: Height above the ellipsoid (C{meter}, same units as this triaxial's C{a}, C{b} and C{c} axes). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. ''' return self._forward3(height, name, *sincos2d_(lat, lon)) def forwardLatLon_(self, slat, clat, slon, clon, height=0, name=NN): '''Convert I{geodetic} lat-, longitude and heigth to cartesian. @arg slat: Geodetic latitude C{sin(lat)} (C{scalar}). @arg clat: Geodetic latitude C{cos(lat)} (C{scalar}). @arg slon: Geodetic longitude C{sin(lon)} (C{scalar}). @arg clon: Geodetic longitude C{cos(lon)} (C{scalar}). @arg height: Height above the ellipsoid (C{meter}, same units as this triaxial's axes C{a}, C{b} and C{c}). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. ''' sa, ca = self._norm2(slat, clat) sb, cb = self._norm2(slon, clon) return self._forward3(height, name, sa, ca, sb, cb) def _forward3(self, h, name, sa, ca, sb, cb): '''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}. ''' ca_x_sb = ca * sb # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 t = fsum_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2, floats=True) N = self.a / self._sqrt(t) # prime vertical x = (h + N) * ca * cb y = (h + N * self._1e2ab) * ca_x_sb z = (h + N * self._1e2ac) * sa return Vector3Tuple(x, y, z, name=name) @Property_RO def _k2_kp2(self): '''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}. ''' # k2 = a2b2 / a2c2 * c2_b2 # kp2 = b2c2 / a2c2 * a2_b2 # b2 = b**2 # xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2) # yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2) # aE = Elliptic(kp2, 0, k2, 1) return (self._a2b2 / self._a2c2 * self._c2_b2, self._b2c2 / self._a2c2 * self._a2_b2) def _radialTo3(self, sbeta, cbeta, somega, comega): '''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega} to cartesian coordinates I{on the triaxial's surface}, also I{ordered} helper for C{.height4}. ''' sa, ca = self._norm2(sbeta, cbeta) sb, cb = self._norm2(somega, comega) b2_a2 = self._1e2ab # == (b/a)**2 c2_a2 = -self._1e2ac # == -(c/a)**2 a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2 x = Fsum(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2) z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) x = self.a * cb * self._sqrt(x) y = self.b * ca * sb z = self.c * sa * self._sqrt(z) return x, y, z def reverseBetaOmega(self, x, y, z, name=NN): '''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega} and height. @arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). @arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). @kwarg name: Optional name (C{str}). @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and C{omega} in C{Radians} and (radial) C{height} in C{meter}, same units as this triaxial's axes. @see: Methods L{Triaxial.forwardBetaOmega} and L{Triaxial.forwardBetaOmega_} and U{Expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/ sites/111/2021/12/09_Panou.pdf>}. ''' v = Vector3d(x, y, z) a, b, h = self._reverse3(x, y, z, atan2, v, self.forwardBetaOmega_) return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, name=name) def reverseCartesian(self, x, y, z, h, normal=True, eps=_EPS2e4, name=NN): '''"Unproject" a cartesian on to a cartesion off this triaxial's surface. @arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). @arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). @arg h: Height above or below this triaxial's surface (C{meter}, same units as B{C{x}}). @kwarg normal: If C{True} the height is C{normal} to the surface, otherwise C{radially} to the center of this triaxial (C{bool}). @kwarg eps: Tolerance for surface test (C{scalar}). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}. ''' v = Vector3d(x, y, z, name=name) s = self.sideOf(v.x, v.y, v.z, eps=eps) if s: # PYCHOK no cover t = _SPACE_((_inside_ if s < 0 else _outside_), self) raise TriaxialError(eps=eps, sideOf=s, x=x, y=y, z=z, txt=t) if h: if normal: v = v.plus(self.normal3d(v.x, v.y, v.z, length=h)) elif v.length > EPS0: v = v.times(_1_0 + (h / v.length)) return v.xyz # Vector3Tuple def reverseLatLon(self, x, y, z, name=NN): '''Convert cartesian to I{geodetic} lat-, longitude and height. @arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). @arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). @kwarg name: Optional name (C{str}). @return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon} in C{degrees} and (radial) C{height} in C{meter}, same units as this triaxial's axes. @see: Methods L{Triaxial.forwardLatLon} and L{Triaxial.forwardLatLon_} and U{Expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/ sites/111/2021/12/09_Panou.pdf>}. ''' v = Vector3d(x, y, z) x *= self._1e2ac # == 1 - e_sub_x**2 y *= self._1e2bc # == 1 - e_sub_y**2 t = self._reverse3(x, y, z, atan2d, v, self.forwardLatLon_) return LatLon3Tuple(*t, name=name) def _reverse3(self, x, y, z, atan2_, v, forward_): '''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}. ''' d = hypot( x, y) a = atan2_(z, d) b = atan2_(y, x) h = v.minus_(*forward_(z, d, y, x)).length return a, b, h class JacobiConformal(Triaxial): '''This is a conformal projection of a triaxial ellipsoid to a plane in which the C{X} and C{Y} grid lines are straight. Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are returned in the case of an ellipsoid of revolution (or a sphere, I{currently not supported}). Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2020) and licensed under the MIT/X11 License. @note: This constructor can not be used to specify a sphere. @see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/ C++/doc/classGeographicLib_1_1JacobiConformal.html#details>}, U{Jacobi's conformal projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>} and Jacobi, C. G. J. I{U{Vorlesungen über Dynamik<https://Books.Google.com/books? id=ryEOAAAAQAAJ&pg=PA212>}}, page 212ff, ''' # @Property_RO # def ab(self): # '''Get relative magnitude C{ab} (C{None} or C{meter}, same units as B{C{a}}). # ''' # return self._ab # @Property_RO # def bc(self): # '''Get relative magnitude C{bc} (C{None} or C{meter}, same units as B{C{a}}). # ''' # return self._bc @Property_RO def _xE(self): '''(INTERNAL) Get the x-elliptic function. ''' k2, kp2 = self._k2_kp2 # -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2 return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2) def xR(self, omega): '''Compute a Jacobi Conformal C{x} projection. @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). @return: The C{x} projection (C{Radians}). ''' return self.xR_(*_SinCos2(omega)) def xR_(self, somega, comega): '''Compute a Jacobi Conformal C{x} projection. @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). @return: The C{x} projection (C{Radians}). ''' s, c = self._norm2(somega, comega, self.a) return Radians(x=self._xE.fPi(s, c) * self._a2_b2) @Property_RO def xyQ2(self): '''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}). ''' return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi), Radians(y=self._c2_b2 * self._yE.cPi), name=JacobiConformal.xyQ2.name) def xyR2(self, beta, omega, name=NN): '''Compute a Jacobi Conformal C{x} and C{y} projection. @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). @kwarg name: Optional name (C{str}). @return: A L{Jacobi2Tuple}C{(x, y)}. ''' return self.xyR2_(*(_SinCos2(beta) + _SinCos2(omega)), name=name or self.xyR2.__name__) def xyR2_(self, sbeta, cbeta, somega, comega, name=NN): '''Compute a Jacobi Conformal C{x} and C{y} projection. @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). @kwarg name: Optional name (C{str}). @return: A L{Jacobi2Tuple}C{(x, y)}. ''' return Jacobi2Tuple(self.xR_(somega, comega), self.yR_(sbeta, cbeta), name=name or self.xyR2_.__name__) @Property_RO def _yE(self): '''(INTERNAL) Get the x-elliptic function. ''' kp2, k2 = self._k2_kp2 # swapped! # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) def yR(self, beta): '''Compute a Jacobi Conformal C{y} projection. @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). @return: The C{y} projection (C{Radians}). ''' return self.yR_(*_SinCos2(beta)) def yR_(self, sbeta, cbeta): '''Compute a Jacobi Conformal C{y} projection. @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). @return: The C{y} projection (C{Radians}). ''' s, c = self._norm2(sbeta, cbeta, self.c) return Radians(y=self._yE.fPi(s, c) * self._c2_b2) class TriaxialError(_ValueError): '''Raised for L{Triaxial} issues. ''' pass # ... def _hartzell3d2(pov, los, abc3, Error=TriaxialError): # MCCABE 13 in .formy.hartzell '''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ...", I{unordered}. ''' def _order2(a, b, c): # @return: 2-Tuple ((a, b, c), un) ordered a >= b >= c if a < b or b < c: t = (a, 0), (b, 1), (c, 2) return _zip(*reversed(sorted(t))) # strict=True else: # a >= b >= c return (a, b, c), None # or () def _order3d(un, v): # @return: Vector3d(x, y, z) un/ordered if un: _, xyz = _zip(*sorted(_zip(un, v.xyz))) # strict=True v = Vector3d(*xyz) return v (a, b, c), un = _order2(*abc3) if not (isfinite(a) and c > 0): raise Error(_invalid_) a2 = a**2 # largest, factored out b2, p2 = (b**2, (b / a)**2) if a != b else (a2, _1_0) c2, q2 = ((c**2, (c / a)**2) if a != c else (a2, _1_0)) if b != c else (b2, _1_0) p3 = _order3d(un, _otherV3d(pov=pov)) u3 = _order3d(un, _otherV3d(los=los)) if los else p3.negate() u3 = u3.unit() # unit vector, opposing signs x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz ux, vy, wz = u3.times_(p3).xyz u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz t = (p2 * c2), c2, b2 m = fdot(t, u2, v2, w2) # a2 factored out if m < EPS0: # zero or near-null LOS vector raise Error(_near_(_null_)) r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, -w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, -w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) if r > 0: # a2 factored out r = sqrt(r) * b * c # == a * a * b * c / a2 elif r < 0: # LOS pointing away from or missing the triaxial raise Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out if d > 0: # POV inside or LOS missing, outside the triaxial raise Error(_inside_ if min(x2 - a2, y2 - b2, z2 - c2) < EPS else _outside_) elif fsum_(x2, y2, z2, floats=True) < d**2: # d past triaxial's center raise Error(_too_(_distant_)) v = p3.minus(u3.times(d)) # Vector3d h = p3.minus(v).length # distance to triaxial return _order3d(un, v), h E = _WGS84.ellipsoid _WGS84 = Triaxial(E.a + 35, E.a - 35, E.b, name=E.name + '+/-35') del E def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN): '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight from a Point-Of-View in space. @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} or L{Vector3d}). @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or C{None} to point to the tri-/biaxial's center. @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal}) or biaxial ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius in C{meter}). @kwarg name: Optional name (C{str}). @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} the distance from B{C{pov}} to C{(x, y, z)} along B{C{los}}. @raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside the tri-/biaxial or B{C{los}} points outside the tri-/biaxial or points in an opposite direction. @raise TypeError: Invalid B{C{pov}} or B{C{los}}. @see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. ''' def _Error(txt): return TriaxialError(pov=pov, los=los, tri_biax=tri_biax, txt=txt) if isinstance(tri_biax, Triaxial_): T = tri_biax else: D = tri_biax if isinstance(tri_biax, Datum) else \ _spherical_datum(tri_biax, name=hartzell4.__name__) E = D.ellipsoid T = Triaxial_(E.a, E.a, E.b, name=E.name) v, h = _hartzell3d2(pov, los, T._abc3, _Error) return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__) def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 13 '''(INTERNAL) Nearest point on and distance to an I{ordered} 2-D ellipse. @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/ Documentation/DistancePointEllipseEllipsoid.pdf>}. ''' def _root2d(r, u, v, g, eps): # robust root finder _1, __2 = _1_0, _0_5 _a, _h2 = fabs, hypot2 u *= r t0 = v - _1 t1 = _0_0 if g < 0 else (hypot(u, v) - _1) for i in range(1, _TRIPS): t = (t0 + t1) * __2 e = _a(t0 - t1) if t in (t0, t1) or e < eps: break g = _h2(u / (t + r), v / (t + _1)) - _1 if g > 0: t0 = t elif g < 0: t1 = t else: break else: # PYCHOK no cover t = _root2d.__name__ raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) return t, i if not (a >= b > 0): raise TriaxialError(a=a, b=b, txt=_not_ordered_) i = None if y: if x: u = fabs(x / a) v = fabs(y / b) g = hypot2(u, v) - _1_0 if g: r = (a / b)**2 t, i = _root2d(r, u, v, g, eps) a = x / (t / r + _1_0) b = y / (t + _1_0) d = hypot(x - a, y - b) else: # on the ellipse a, b, d = x, y, _0_0 else: # x == 0 if y < 0: b = -b a, d = x, fabs(y - b) else: # y == 0 n = a * x d = (a + b) * (a - b) if d > fabs(n): # PYCHOK no cover r = n / d a *= r b *= sqrt(_1_0 - r**2) d = hypot(x - a, b) else: if x < 0: a = -a b, d = y, fabs(x - a) return a, b, d, i def _normalTo4u(x, y, a, b, eps=EPS): # PYCHOK no cover '''(INTERNAL) Nearest point on and distance to an I{unordered} 2-D ellipse. ''' if a < b: b, a, d, i = _normalTo4(b, a, y, x, eps=eps) else: a, b, d, i = _normalTo4(a, b, x, y, eps=eps) return a, b, d, i def _normalTo5(x, y, z, T, eps=EPS): # MCCABE 23 '''(INTERNAL) Nearest point on and distance to an I{ordered} triaxial. @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. ''' def _root3d(r, s, u, v, w, g, eps): # robust root finder _1, __2 = _1_0, _0_5 _a, _h2 = fabs, hypot2_ u *= r v *= s t0 = w - _1 t1 = _0_0 if g < 0 else (hypot_(u, v, w) - _1) for i in range(1, _TRIPS): t = (t0 + t1) * __2 e = _a(t0 - t1) if t in (t0, t1) or e < eps: break g = _h2(u / (t + r), v / (t + s), w / (t + _1)) - _1 if g > 0: t0 = t elif g < 0: t1 = t else: break else: # PYCHOK no cover t = _root3d.__name__ raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) return t, i a, b, c = T._abc3 if not (a >= b >= c > 0): raise TriaxialError(a=a, b=b, c=c, triaxial=T, txt=_not_ordered_) if eps > 0: val = max(eps * 1e8, EPS) else: # no validation val, eps = 0, -eps i = None if z: if y: if x: u = fabs(x / a) v = fabs(y / b) w = fabs(z / c) g = hypot2_(u, v, w) - _1_0 if g: r = T._1e2ac # (c / a)**2 s = T._1e2bc # (c / b)**2 t, i = _root3d(_1_0 / r, _1_0 / s, u, v, w, g, eps) a = x / (t * r + _1_0) b = y / (t * s + _1_0) c = z / (t + _1_0) d = hypot_(x - a, y - b, z - c) else: # on the ellipsoid a, b, c, d = x, y, z, _0_0 else: # x == 0 a = x # 0 b, c, d, i = _normalTo4(y, z, b, c, eps=eps) elif x: # y == 0 b = y # 0 a, c, d, i = _normalTo4(x, z, a, c, eps=eps) else: # x == y == 0 if z < 0: c = -c a, b, d = x, y, fabs(z - c) else: # z == 0 t = False n = a * x d = T._a2c2 # (a + c) * (a - c) if d > fabs(n): u = n / d n = b * y d = T._b2c2 # (b + c) * (b - c) if d > fabs(n): v = n / d d = _1_0 - hypot2(u, v) if d > 0: a *= u b *= v c *= sqrt(d) d = hypot_(x - a, y - b, c) t = True if not t: c = z # 0 a, b, d, i = _normalTo4(x, y, a, b, eps=eps) if val > 0: # validate e = T.sideOf(a, b, c, eps=val) if e: # not near the ellipsoid's surface raise TriaxialError(a=a, b=b, c=c, d=d, x=x, y=y, z=z, eps=val, sideOf=e, txt=T.toRepr()) if d: # angle of delta and normal vector m = Vector3d(x, y, z).minus_(a, b, c) if m.euclid > val: m = m.unit() n = T.normal3d(a, b, c) e = n.dot(m) # n.negate().dot(m) if not isnear1(fabs(e), eps1=val): raise TriaxialError(n=n, m=m, eps=val, dot=e, txt=T.toRepr()) return a, b, c, d, i def _normalTo5u(x, y, z, a, b, c, eps=EPS): '''(INTERNAL) Nearest point on and distance to an I{unordered} 3-D, triaxial ellipsoid. ''' def _order3(a, b, c, x, y, z): # @return: 3-Tuple ((a, b, c), un, (x, y, z)), ordered a >= b >= c t = (a, 0, x), (b, 1, y), (c, 2, z) return _zip(*reversed(sorted(t))) # strict=True def _unorder5(un, abcdi): # @return: 5-Tuple (a, b, c, d, i), unordered un += 3, 4 # keep d and i in place return tuple(_zip(*sorted(_zip(un, abcdi))))[1] # strict=True abc, un, xyzT = _order3(a, b, c, x, y, z) xyzT += Triaxial(*abc), return _unorder5(un, _normalTo5(*xyzT, eps=eps)) def _SinCos2(x): '''Get C{sin} and C{cos} of C{x} in C{Degrees}, C{Radians} or {radians}. ''' return sincos2d(x) if isinstance(x, Degrees) else ( sincos2(x) if isinstance(x, Radians) else sincos2(float(x))) # assume C{radians} if __name__ == '__main__': from pygeodesy import printf # <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km # Planet # <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> # <https://link.Springer.com/article/10.1007/s00190-022-01650-9> for n, a, b, c in (('Amalthea', 125.0, 73.0, 64), # Jupiter ('Ariel', 581.1, 577.9, 577.7), # Uranus ('Earth', 6378.173435, 6378.1039, 6356.7544), ('Enceladus', 256.6, 251.4, 248.3), # Saturn ('Europa', 1564.13, 1561.23, 1560.93), # Jupiter ('Io', 1829.4, 1819.3, 1815.7), # Jupiter ('Mars', 3394.6, 3393.3, 3376.3), ('Mimas', 207.4, 196.8, 190.6), # Saturn ('Miranda', 240.4, 234.2, 232.9), # Uranus ('Moon', 1735.55, 1735.324, 1734.898), # Earth ('Tethys', 535.6, 528.2, 525.8)): # Saturn t = Triaxial(km2m(a), km2m(b), km2m(c), name=n) printf('# %r', t) if n == 'Earth': printf('# %r', JacobiConformal(t.a, t.b, t.c, name=n)) printf('# %r', _WGS84) # **) MIT License # # Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. # % python3 -m pygeodesy.triaxials # Triaxial(name='Amalthea', a=125000, b=73000, c=64000, e2ab=0.658944, e2bc=0.231375493, e2ac=0.737856, volume=2446253479595252, area=93239507787.490371704, area_p=93212299402.670425415) # Triaxial(name='Ariel', a=581100, b=577900, c=577700, e2ab=0.01098327, e2bc=0.000692042, e2ac=0.011667711, volume=812633172614203904, area=4211301462766.580078125, area_p=4211301574065.829589844) # Triaxial(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, volume=1083208241574987694080, area=510065911057441.0625, area_p=510065915922713.6875) # JacobiConformal(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, xyQ2=xyQ2(x=1.572084, y=4.249876), volume=1083208241574987694080, area=510065911057441.0625, area_p=510065915922713.6875) # Triaxial(name='Enceladus', a=256600, b=251400, c=248300, e2ab=0.040119337, e2bc=0.024509841, e2ac=0.06364586, volume=67094551514082248, area=798618496278.596679688, area_p=798619018175.109863281) # Triaxial(name='Europa', a=1564130, b=1561230, c=1560930, e2ab=0.003704694, e2bc=0.000384275, e2ac=0.004087546, volume=15966575194402123776, area=30663773697323.51953125, area_p=30663773794562.45703125) # Triaxial(name='Io', a=1829400, b=1819300, c=1815700, e2ab=0.011011391, e2bc=0.003953651, e2ac=0.014921506, volume=25313121117889765376, area=41691875849096.7421875, area_p=41691877397441.2109375) # Triaxial(name='Mars', a=3394600, b=3393300, c=3376300, e2ab=0.000765776, e2bc=0.009994646, e2ac=0.010752768, volume=162907283585817247744, area=144249140795107.4375, area_p=144249144150662.15625) # Triaxial(name='Mimas', a=207400, b=196800, c=190600, e2ab=0.09960581, e2bc=0.062015624, e2ac=0.155444317, volume=32587072869017956, area=493855762247.691894531, area_p=493857714107.9375) # Triaxial(name='Miranda', a=240400, b=234200, c=232900, e2ab=0.050915557, e2bc=0.011070811, e2ac=0.061422691, volume=54926187094835456, area=698880863325.756958008, area_p=698881306767.950317383) # Triaxial(name='Moon', a=1735550, b=1735324, c=1734898, e2ab=0.000260419, e2bc=0.000490914, e2ac=0.000751206, volume=21886698675223740416, area=37838824729886.09375, area_p=37838824733332.2265625) # Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, volume=623086233855821440, area=3528073490771.394042969, area_p=3528074261832.738769531) # Triaxial(name='WGS84+/-35', a=6378172, b=6378102, c=6356752.314245179, e2ab=0.00002195, e2bc=0.006683478, e2ac=0.006705281, volume=1083207319768789942272, area=510065621722018.125, area_p=510065626587483.3125)